1. Principles of Statistics and Economics Statistics SHUFE
Dongdong Ge
Principles of Statistics and Economics Statistics SHUFE
Instructor: Dongdong Ge

2. Four things about probability distribution
Applied situation (适用情境)
Probability function
Calculate the probability for a well-defined event…
Expected value and variance
Hmk: chap
chap
chap

3. Learning Objectives Continuous random variable
Uniform Distribution (均匀分布)
Normal Distribution (正态分布) and Z-score
Exponential Distribution (指数分布)

4. Example
Consider a random variable x – represent the flight time of an airplane traveling from Chicago to New York. It could range from minutes.
Difference from the discrete random variable
Can be any value in the interval from minutes.
It is not meaningful to talk about probability at a particular value.

5. Uniform Probability Distribution
Probability Density Function
(概率密度函数)

6. Uniform Probability Distribution
Calculate the probability for a well-defined event…
P(X1≤X ≤ X2) can be found by computing the area under the graph of f(x) over the interval from X1 to X2 .
Expected value:
Variance:
Note the error in the textbook!

7. Normal Probability Distribution (正态分布)
Many natural phenomena follow a normal distribution or something very close to it
Example: driving times to work, weight of a population, IQ score
Widely used in inferential statistics, which is the major topic for the rest of the semester.

8. Normal distribution Takes a bell-shaped normal cure.
Probability density function
Expected value
Standard deviation
Takes a bell-shaped
normal cure.

9. Normal distribution
Is differentiated by its mean and standard deviation;
The highest point is at the mean, which is also the median and mode;
The mean of the distribution could be any value.

10. Normal Distribution Symmetrical around the mean ;
The total area under the curve = probability for all possible values = 1.
We use the distribution to get the probability that the variable will lie in a certain range of values.

11. Characteristics of Normal Distribution
Note that these values are approximations. For example, according to the normal curve probability density function, 95% of the data will fall within 1.96 standard deviations of the mean; 2 standard deviations is a convenient approximation.
68% of the data will fall within 1 standard deviation of the mean
95% of the data will fall within 2 (1.96) standard deviations of the mean
Almost all (99.7%) of the data will fall within 3 standard deviations of the mean

12. Standard Normal Distribution （标准正态分布）
In Standard normal distribution;
μ = 0 and σ = 1
The letter z is commonly used to designate this particular normal random variables.
Use Z-score and Normal Distribution Table to calculate the probability

13. Exercise P(0.00≤z≤1.25) P(-1.25≤z≤1.25) P(-1.05≤z≤1.25) P(z≥1.25)
Find a z value such that the probability of obtaining a larger z value is .10.

14. Convert Normal Distribution To Standard Normal Distribution
(P<52)
Z=(52-100)/16 = -3
P(z<-3) = = 1 52
100
116

15. Calculate probabilities in a Normal Distribution
Use Z-score and Normal Distribution Table
Where,
xi = original value
μ = mean
σ = standard deviation

16. Taxi time from Jiaoda to Pudong Airport:  = 54 minutes,  = 4
Taxi time from Jiaoda to Pudong Airport:  = 54 minutes,  = 4.6 minutes.
P(55 minutes) =
P(<48 minutes) =
P(>53 minutes) =
P(between 55 and 60) =
P(between 52 and 65) =
The lowest 5% of travel times are below how many minutes?

17. Taxi time from Jiaoda to Pudong Airport:  = 54 minutes,  = 4
Taxi time from Jiaoda to Pudong Airport:  = 54 minutes,  = 4.6 minutes.
P(55 minutes): Z = (55-54)/4.6=.22; P(Z=.22)= 0
P(<48 minutes): Z= (48-54)/4.6 = -1.30; P(Z<-1.30) = = .0968
P(>53 minutes): Z = (53-54)/4.6= -.2; P(Z>-.22) = = .5871
P(between 55 and 60); Z=(60-54)/4.6= 1.30; P(0<Z<1.30) = .4032
P(between 52 and 65); Z = (52-54)/4.6 = -0.44; Z= (65-54)/4.6= 2.39; P(-0.44<Z<2.39)= = .662
The lowest 5% of travel times are below how many minutes? P(Z<A)=.05 ; (x-54)/4.6 ≈ ; x ≈

18. Normal Approximation of Binomial Probability
# of successes in a binomial experiment can be approached by a normal distribution if:
In this case,

19. Example 10% error, 100 invoices. The probability of 12 errors?
Binomial distribution:
Normal approx:
P(11.5<=x<=12.5)
.5:continuity correction factor
Z=( )/3=.83
Z=( )/3=.50
P= (diff=0.0064)

20. Exponential Probability Distribution
Used for random variables the time between customers at a clothing shop; the time required for a truck load etc…
Difference from Poisson distribution:
Poisson describes the number of occurrences per interval;
Exponential describes the length of the interval between occurrences.

21. Exponential Probability Distribution
Probability density function
Calculate probability
Expected value= Standard deviation

22. Exponential Probability Distribution

23. Relationship between the Poisson and Exponential distributions
Possion:
Expeonetial:
Expect 10 cars in one hour. Then: